Problem 41 - A Graphical Approach to Precalculus with Limits

Question

Determine each limit, if it exists. $$\lim _{x \rightarrow 1}[\sqrt{x}(1+x)]$$

Step-by-Step Solution

Answer

The limit is 2.

Step 1: Plug in the value of x

To find \( \lim_{x \to 1} \sqrt{x}(1+x) \), start by plugging the value \( x = 1 \) directly into the expression.

Step 2: Evaluate the square root term

When \( x = 1 \), \( \sqrt{x} = \sqrt{1} = 1 \). So, the expression simplifies to \( 1 \times (1+1) \).

Step 3: Simplify the expression

Now, simplify the expression by performing the addition: \( 1 + 1 = 2 \). Thus, the expression \( 1 \times 2 \) simplifies to \( 2 \).

Step 4: Conclusion

Since we were able to directly substitute the value of \( x = 1 \) and obtain a finite result, the limit is \( 2 \).

Key Concepts

Direct Substitution

In calculus, evaluating limits by direct substitution is often the simplest and most straightforward method. It involves plugging the value that the variable approaches directly into the expression. This technique works well in cases where the function is continuous at that point.

• Check for continuity: Direct substitution is only valid if the function is continuous at the point in question. For this to happen, there should be no holes, jumps, or vertical asymptotes in the graph of the function at that point.
• If the function is continuous, plug in the value: Simply replace the variable with the value it is approaching to find the limit.
• Check the result: If the substitution yields a finite and defined number, then that number is the limit.

Direct substitution is not always possible if the function has a discontinuity at the point. However, in this example, as we substitute \( x = 1 \) into \( \sqrt{x}(1+x) \), we end up simply solving the expression \( \sqrt{1}(1 + 1) = 2 \). This demonstrates a straightforward example of using direct substitution.

Square Root Evaluation

Evaluating square roots is an essential skill in algebra and calculus when simplifying expressions. In the context of limits, understanding how to simplify expressions that include square roots is important for evaluating them smoothly.

• Identify the term: Recognize the part of the expression that contains a square root.
• Simplify the square root: Calculate the square root of the number or variable provided. For example, \( \sqrt{1} = 1 \).
• Proceed with simplification: Once the square root is evaluated, use it to further simplify the overall expression.

In the problem at hand, the square root alone was evaluated by simply recognizing that \( \sqrt{1} = 1 \). This simplification made it easier to handle the remaining parts of the expression \( 1 \times (1 + 1) \). Understanding square roots as literal representations of multiplying a number by itself provides clarity, allowing for straightforward calculations in limit evaluations.

Evaluating Limits

Evaluating limits is a fundamental concept in calculus that aids in understanding the behavior of functions as they approach particular points. It's a crucial step for derivatives and integrals as well.

• Understand the approach: A limit asks us to find the value that a function approaches as the variable gets infinitely close to a certain number.
• Utilize methods: Techniques include direct substitution, factoring, rationalizing, or using L'Hopital's Rule if the substitution method directly yields an indeterminate form like \(0/0\).
• Confirm the result: Check the limit algebraically and graphically if possible, to ensure consistency.

In the given problem, the limit was determined using direct substitution since it was possible to simplify the expression without indeterminacies. By methodically substituting \( x = 1 \) into \( \sqrt{x}(1 + x) \) and simplifying, it was confirmed that the limit is \( 2 \), offering a neat conclusion to the problem's evaluation.